The ratio of boys to girls in a class is $5:4$ . If a few boys leave the class, the ratio of boys to girls would become the reciprocal of the earlier ratio, which of the following CANNOT be the possible number of boys and girls in the class? $\newline$ A) $45$ $\newline$ B) $90$ $\newline$ C) $120$ $\newline$ D) $135$

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Q. The ratio of boys to girls in a class is

5:4

. If a few boys leave the class, the ratio of boys to girls would become the reciprocal of the earlier ratio, which of the following CANNOT be the possible number of boys and girls in the class?

\newline

45

\newline

90

\newline

120

\newline

135

Original Ratio Setup: The original ratio of boys to girls is $5:4$ . If we let the number of boys be $5x$ and the number of girls be $4x$ , where $x$ is a common multiplier.
Reciprocal Ratio Setup: When some boys leave, the ratio becomes $4:5$ , which is the reciprocal of the original ratio. So, the new number of boys must be $4x$ and the number of girls must be $5x$ .
Equation Setup: Since the number of girls doesn't change, only boys are leaving, we can set up the equation $5x - \text{(number of boys that leave)} = 4x$ .
Solving for Boys Leaving: Solving the equation, we get that the number of boys that leave is $x$ . This means that the total number of students, boys plus girls, is $5x + 4x = 9x$ .
Checking Total Number: The total number of students must be a multiple of $9$ . We need to check which of the given options is not a multiple of $9$ .
Option Analysis: Option A: $45$ is divisible by $9$ ( $45 \div 9 = 5$ ), so it could be the total number of students.
Option Analysis: Option A: $45$ is divisible by $9$ ( $45 \div 9 = 5$ ), so it could be the total number of students.Option B: $90$ is divisible by $9$ ( $90 \div 9 = 10$ ), so it could be the total number of students.
Option Analysis: Option A: $45$ is divisible by $9$ ( $45 \div 9 = 5$ ), so it could be the total number of students.Option B: $90$ is divisible by $9$ ( $90 \div 9 = 10$ ), so it could be the total number of students.Option C: $120$ is divisible by $9$ ( $120 \div 9 = 13.333$ ), which is not a whole number, so it cannot be the total number of students.